The small quantum fluctuations in the scalar field lead to fluctuations in
the energy density which was studied in a perturbative theory [1]}.
As discussed in [2]} quantum fluctuations effect of the inflaton
are generally negligibles, since the coupling of the scalar field to bulk
gravitational fluctuations only modifies the usual 4D predictions at the
next order in the slow-roll expansion. So, one can define the power spectrum
of the curvature perturbations as \(P_{R}\left( k\right) =\left( \frac{H^{2}}{2\pi \overset{\cdot }{\phi }}\right) ^{2}\) . On the other hand, the quantum
fluctuations in the scalar field lead also to fluctuations in the metric
[3]}. In this way, one can define the amplitude of tensor
perturbations as \(P_{g}\left( k\right) =\frac{8}{M_{p}^{2}}\left( \frac{H}{2\pi }\right) ^{2}F^{2}\left( x\right) .\)  These results lead to the ratio of
tensor to scalar perturbations \(r=\frac{P_{g}\left( k\right) }{P_{R}\left(k\right) }.\)  In relation to \(P_{R}\left( k\right) \) , the scalar spectral
index is defined as \(n_{s}-1=\frac{d\ln P_{R}\left( k\right) }{d\ln \left(k\right) }.\)  Refs. [4]} are recommended for further reading on
inflation.
